How to draw a Cartesian plane like this one?

Question

I read some topics on the forum, but still have no idea on how to draw a diagram like this one

I know I could just insert the image, but I think it would be nicer to draw it myself, and I also have to change the variables (I need to change S with T, X with J and J with X)

This is what I managed to achieve so far

\documentclass{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.5}
\begin{document}
\begin{tikzpicture}
\begin{axis}[axis lines=middle, axis equal, title=test, xlabel={Time}, ylabel=.{States}]
\addplot table {
x y
1.7 4.3
0 4.3

1.7 4
2.3 4

2.3 5.2
3.1 5.2

3.1 4.5
3.8 4.5

3.8 3.8
4.7 3.8

4.7 5.5
5.8 5.5
};
\end{axis}
\end{tikzpicture}
\end{document}

I still need the ticks on the "x" axis to be related to the jump points of the function, and to name them "T_0, T_1, ... T_n" (note that the jump length is random, and it has to be). Moreover, does anyone know how to remove the right dot from the function, leaving only the left one?

P.S: Of course I will add reference on the bibliography for using this image

A big hug to everybody!

Answer 1

This is probably not the most flexible solution, but it's fairly straight forward I think. (You may well get better solutions from others later.) It also doesn't use pgfplots, just plain TikZ.

\documentclass[border=5mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{arrows.meta}
\begin{document}
\begin{tikzpicture}[
   axis/.style={thick,Latex-Latex},
   xbar/.style={-Circle}
]

\draw [axis] (0,4) node[above] {States} |- (7,0) node[below] {time};

\foreach \y/\txt in {1/J_n=k,2/J_0=i,2.5/J_1=j}
  \node [left=3mm] at (0,\y) {$\{\txt\}$};

\foreach [count=\i] \x/\txt in {0/0,1/1,1.8/2,3.2/n,5/n+1}
{
   \node [below=3mm] (S\i) at (\x,0) {$S_{\txt}$};
   \draw (\x,0) -- ++(0,{ifthenelse(\x>0,-3pt,0)});
}

\foreach \xA/\xB/\y in {0/1/2,1/1.8/2.5,3.2/5/1}
  \draw [xbar] (\xA,\y) -- (\xB,\y);

\path (S3) -- node (dots1) {$\dots$} (S4);
\node [right=2mm] (dots2) at (S5.east) {$\dots$}; 
\node at (0,1.5 -| dots1) {$\dots$};
\node at (0,1 -| dots2) {$\dots$};

\node [below left] at (7,4) {
\begin{tabular}{c @{ : }l}
$(X_n)$ & sojourn time \\
$(J_n)$ & states of the system \\
$(S_n)$ & jump time
\end{tabular}};

\end{tikzpicture}
\end{document}